Cole, 20, was named in Sven-Goran Eriksson's World Cup squad on Thursday despite having only four England caps.
Gascoigne went into the 1990 World Cup as a relative unknown on the international stage, but left it as one of the stars of the tournament.
And he thinks that Cole can also make a name for himself this summer.
"He has got a frightening talent - I think he might be a bit better than I was," Gascoigne said.
"You want someone who can come on in the second-half and offer you something different and Joe can do that.
"His self-belief is unbelievable. If you've got a lot of confidence in yourself, you're halfway there."
Cole said: "I am obviously delighted. I have no doubts about my ability and I am sure that I can do a job for the team if selected to play.
"It will be great experience for me.
"I would like to acknowledge the help and advice given to me by Glenn Roeder and everyone at West Ham and to family and friends for support.
"With this being my first World Cup it's going to be a tremendous thrill for me, and hopefully, if it goes well, the highlight of my career so far."
One of the first people to congratulate Cole was the Queen, who visited West Ham during a Golden Jubilee tour of east London.
Leeds defender Danny Mills, another to be included, is confident he can fill the void left by the injured Gary Neville at right-back.
"I'd like to think I can play as well as anyone in Gary's spot," Mills said.
I thought my chances of making the squad had
gone back then - I was desperate
West Ham keeper
"It's my preferred position and if I do get the nod from Mr Eriksson I'm confident I won't let him down."
David James was relieved to be named in the squad after sustaining a serious knee injury just nine months ago.
The West Ham keeper collided with Martin Keown just minutes after coming on as a substitute in a 2-0 friendly defeat by Holland.
James spent just over three months on the sidelines but fought his way back and has been in superb form for the Hammers.
"I thought my chances of making the squad had gone back then - I was desperate," he said.
"The possible diagnosis was anterior cruciate and ligament damage and that would have taken a year after the surgery to get back."
Newcastle midfielder Kieron Dyer, who has made only one appearance for England since Eriksson became coach, said: "I've had a good last few weeks with Newcastle, and now I'm fit and raring to go in Japan and Korea."
Middlesbrough centre-half Gareth Southgate said: "Your whole life, you dream of playing in the World Cup.
"It is the pinnacle of your career. To get the opportunity to go and represent your country is fantastic and I'll relish that.
"We're in a very difficult group, but if we get through that, then anything can happen."